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The least infinite-dimensionality for Frechet spaces is c (Mazur), for metrizable barrelled spaces, b (Saxon and Sanchez Ruiz, 1996). For metrizable spaces with the yet weaker inductive property, it is the dimension Aleph[1] of the space chi spanned by any Aleph[1] scalar sequences of the form [(1, x, x^2, x^3, . . .)]. (A locally convex space is inductive if it is the inductive limit of each increasing covering sequence of subspaces). Indeed, chi is a non-barrelled subspace of the Frechet space omega, where the fundamental theorem of algebra at once proves density, dimension and inductivity. Moreover, if each |x| < 1, geometric series then put chi inside the Banach space [l^1], where the identity principle similarly proves the normable case.
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Tom
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97--101
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Bibliogr. 15 poz.,
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bwmeta1.element.baztech-article-BAT2-0001-0909